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Fractional quiver W-algebras

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Abstract

We introduce quiver gauge theory associated with the non-simply laced type fractional quiver and define fractional quiver W-algebras by using construction of Kimura and Pestun (Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1072-1; Lett Math Phys, 2018. https://doi.org/10.1007/s11005-018-1073-0) with representation of fractional quivers.

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Notes

  1. The M-theory brane picture for A-series is rotated by 90\(^{\circ }\).

References

  1. Kimura, T., Pestun, V.: Quiver W-algebras. Lett. Math. Phys. (2018). https://doi.org/10.1007/s11005-018-1072-1. arXiv:1512.08533 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  2. Kimura, T., Pestun, V.: Quiver elliptic W-algebras. Lett. Math. Phys (2018). https://doi.org/10.1007/s11005-018-1073-0. arXiv:1608.04651 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  3. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg–Witten exact solution. Phys. Lett. B 355, 466–474 (1995). arXiv:hep-th/9505035 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  4. Martinec, E.J., Warner, N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B 459, 97–112 (1996). arXiv:hep-th/9509161 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  5. Donagi, R., Witten, E.: Supersymmetric Yang–Mills theory and integrable systems. Nucl. Phys. B 460, 299–334 (1996). arXiv:hep-th/9510101 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  6. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. In: XVIth International Congress on Mathematical Physics, pp. 265–289 (2009). arXiv:0908.4052 [hep-th]

  7. Nekrasov, N., Shatashvili, S.: Bethe Ansatz and supersymmetric vacua. AIP Conf. Proc. 1134, 154–169 (2009)

    Article  ADS  Google Scholar 

  8. Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  9. Wyllard, N.: \(A_{N-1}\) conformal Toda field theory correlation functions from conformal \(\cal{N} = 2\) \(SU(N)\) quiver gauge theories. JHEP 0911, 002 (2009). arXiv:0907.2189 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  10. Awata, H., Yamada, Y.: Five-dimensional AGT conjecture and the deformed Virasoro algebra. JHEP 1001, 125 (2010). arXiv:0910.4431 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  11. Nekrasov, N., Pestun, V.: Seiberg–Witten geometry of four dimensional \(N=2\) quiver gauge theories. arXiv:1211.2240 [hep-th]

  12. Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories. Commun. Math. Phys. 357, 519–567 (2018). arXiv:1312.6689 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  13. Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and \(qq\)-characters. JHEP 1603, 181 (2016). arXiv:1512.05388 [hep-th]

    Article  ADS  Google Scholar 

  14. Nekrasov, N.: BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem. Adv. Theor. Math. Phys. 21, 503–583 (2017). arXiv:1608.07272 [hep-th]

    Article  MathSciNet  Google Scholar 

  15. Nekrasov, N.: BPS/CFT correspondence III: Gauge Origami partition function and \(qq\)-characters. Commun. Math. Phys. 358, 863–894 (2017). arXiv:1701.00189 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  16. Frenkel, E., Reshetikhin, N.: Quantum affine algebras and deformations of the Virasoro and \(\mathscr {W}\)-algebras. Commun. Math. Phys. 178, 237–264 (1996). arXiv:q-alg/9505025

    Article  ADS  MathSciNet  Google Scholar 

  17. Frenkel, E., Reshetikhin, N.: Deformations of \(\cal{W}\)-algebras associated to simple Lie algebras. Commun. Math. Phys. 197, 1–32 (1998). arXiv:q-alg/9708006 [math.QA]

    MathSciNet  MATH  Google Scholar 

  18. Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantum affine algebras and deformations of \(\cal{W}\)-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics, vol. 248 of Contemporary Mathematics, pp. 163–205. American Mathematical Society (1999). arXiv:math/9810055 [math.QA]

  19. Shiraishi, J., Kubo, H., Wata, H.A., Odake, S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996). arXiv:q-alg/9507034

    Article  ADS  MathSciNet  Google Scholar 

  20. Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum \(\cal{W}_N\) algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401–416 (1996). arXiv:q-alg/9508011 [math.QA]

    Article  ADS  Google Scholar 

  21. Moore, G.W., Nekrasov, N., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000). arXiv:hep-th/9712241

    Article  ADS  MathSciNet  Google Scholar 

  22. Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161

    Article  MathSciNet  Google Scholar 

  23. Feigin, B., Finkelberg, M., Negut, A., Rybnikov, L.: Yangians and cohomology rings of Laumon spaces. Selec. Math. 17, 573–607 (2011). arXiv:0812.4656 [math.AG]

    Article  MathSciNet  Google Scholar 

  24. Finkelberg, M., Rybnikov, L.: Quantization of Drinfeld Zastava in type A. J. Eur. Math. Soc. 16, 235–271 (2014). arXiv:1009.0676 [math.AG]

    Article  MathSciNet  Google Scholar 

  25. Kanno, H., Tachikawa, Y.: Instanton counting with a surface operator and the chain-saw quiver. JHEP 1106, 119 (2011). arXiv:1105.0357 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  26. Kimura, T., Nekrasov, N., Pestun, V.: To appear

  27. Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and Triality). arXiv:1506.04183 [hep-th]

  28. Aganagic, M., Frenkel, E., Okounkov, A.: Quantum \(q\)-Langlands correspondence. arXiv:1701.03146 [hep-th]

  29. Haouzi, N., Kozçaz, C.: The ABCDEFG of little strings. arXiv:1711.11065 [hep-th]

  30. Cremonesi, S., Ferlito, G., Hanany, A., Mekareeya, N.: Coulomb branch and the moduli space of instantons. JHEP 1412, 103 (2014). arXiv:1408.6835 [hep-th]

    Article  ADS  Google Scholar 

  31. Dey, A., Hanany, A., Koroteev, P., Mekareeya, N.: On three-dimensional quiver Gauge theories of type B. JHEP 1709, 067 (2017). arXiv:1612.00810 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  32. Kimura, T.: Matrix model from \(\cal{N} = 2\) orbifold partition function. JHEP 1109, 015 (2011). arXiv:1105.6091 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  33. Kimura, T.: \(\beta \)-ensembles for toric orbifold partition function. Prog. Theor. Phys. 127, 271–285 (2012). arXiv:1109.0004 [hep-th]

    Article  ADS  Google Scholar 

  34. Frenkel, E., Hernandez, D.: Langlands duality for finite-dimensional representations of quantum affine algebras. Lett. Math. Phys. 96, 217–261 (2011). arXiv:0902.0447 [math.QA]

    Article  ADS  MathSciNet  Google Scholar 

  35. Marshakov, A., Nekrasov, N.: Extended Seiberg–Witten theory and integrable hierarchy. JHEP 0701, 104 (2007). arXiv:hep-th/0612019

    Article  ADS  MathSciNet  Google Scholar 

  36. Nakajima, H., Yoshioka, K.: Lectures on instanton counting. CRM Proc. Lect. Notes 38, 31–102 (2003). arXiv:math/0311058 [math.AG]

    Article  MathSciNet  Google Scholar 

  37. Aganagic, M., Haouzi, N., Kozçaz, C., Shakirov, S.: Gauge/Liouville Triality. arXiv:1309.1687 [hep-th]

  38. Aganagic, M., Haouzi, N., Shakirov, S.: \(A_n\)-Triality. arXiv:1403.3657 [hep-th]

  39. Bouwknegt, P., Pilch, K.: On deformed \(\cal{W}\)-algebras and quantum affine algebras. Adv. Theor. Math. Phys. 2, 357–397 (1998). arXiv:math/9801112 [math.QA]

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The work of T.K. was supported in part by Keio Gijuku Academic Development Funds, JSPS Grant-in-Aid for Scientific Research (No. JP17K18090), the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (No. S1511006), JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (No. JP15H05855), and “Discrete Geometric Analysis for Materials Design” (No. JP17H06462). V.P. acknowledges grant RFBR 16-02-01021. The research of V.P. on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (QUASIFT Grant Agreement No. 677368).

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Authors and Affiliations

  1. Keio University, Tokyo, Japan

    Taro Kimura

  2. IHES, Bures-sur-Yvette, France

    Vasily Pestun

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  1. Taro Kimura
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  2. Vasily Pestun
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Correspondence to Taro Kimura.

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Kimura, T., Pestun, V. Fractional quiver W-algebras. Lett Math Phys 108, 2425–2451 (2018). https://doi.org/10.1007/s11005-018-1087-7

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